Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $a \neq 0$. $y = \dfrac{8a - 40}{a^2 - a} \div \dfrac{a + 4}{a^2 + 3a - 4} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{8a - 40}{a^2 - a} \times \dfrac{a^2 + 3a - 4}{a + 4} $ First factor the quadratic. $y = \dfrac{8a - 40}{a^2 - a} \times \dfrac{(a - 1)(a + 4)}{a + 4} $ Then factor out any other terms. $y = \dfrac{8(a - 5)}{a(a - 1)} \times \dfrac{(a - 1)(a + 4)}{a + 4} $ Then multiply the two numerators and multiply the two denominators. $y = \dfrac{ 8(a - 5) \times (a - 1)(a + 4) } { a(a - 1) \times (a + 4) } $ $y = \dfrac{ 8(a - 5)(a - 1)(a + 4)}{ a(a - 1)(a + 4)} $ Notice that $(a + 4)$ and $(a - 1)$ appear in both the numerator and denominator so we can cancel them. $y = \dfrac{ 8(a - 5)\cancel{(a - 1)}(a + 4)}{ a\cancel{(a - 1)}(a + 4)} $ We are dividing by $a - 1$ , so $a - 1 \neq 0$ Therefore, $a \neq 1$ $y = \dfrac{ 8(a - 5)\cancel{(a - 1)}\cancel{(a + 4)}}{ a\cancel{(a - 1)}\cancel{(a + 4)}} $ We are dividing by $a + 4$ , so $a + 4 \neq 0$ Therefore, $a \neq -4$ $y = \dfrac{8(a - 5)}{a} ; \space a \neq 1 ; \space a \neq -4 $